Given $n$ samples of a function $f\colon D\to\mathbb C$ in random points drawn with respect to a measure $\varrho_S$ we develop theoretical analysis of the $L_2(D, \varrho_T)$-approximation error. For a parituclar choice of $\varrho_S$ depending on $\varrho_T$, it is known that the weighted least squares method from finite dimensional function spaces $V_m$, $\dim(V_m) = m < \infty$ has the same error as the best approximation in $V_m$ up to a multiplicative constant when given exact samples with logarithmic oversampling. If the source measure $\varrho_S$ and the target measure $\varrho_T$ differ we are in the domain adaptation setting, a subfield of transfer learning. We model the resulting deterioration of the error in our bounds. Further, for noisy samples, our bounds describe the bias-variance trade off depending on the dimension $m$ of the approximation space $V_m$. All results hold with high probability. For demonstration, we consider functions defined on the $d$-dimensional cube given in unifom random samples. We analyze polynomials, the half-period cosine, and a bounded orthonormal basis of the non-periodic Sobolev space $H_{\mathrm{mix}}^2$. Overcoming numerical issues of this $H_{\text{mix}}^2$ basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results.

翻译：给定从分布$\varrho_S$中随机抽取的函数$f\colon D\to\mathbb C$的$n$个样本，我们开发了$L_2(D, \varrho_T)$逼近误差的理论分析。对于依赖于$\varrho_T$的特定$\varrho_S$选择，已知在精确样本且对数过采样的条件下，来自有限维函数空间$V_m$（$\dim(V_m)=m<\infty$）的加权最小二乘法具有与$V_m$中最佳逼近相同的误差（最多相差一个乘法常数）。当源分布$\varrho_S$与目标分布$\varrho_T$不同时，我们处于域自适应设置中，这是迁移学习的一个子领域。我们在误差界中刻画了由此导致的误差恶化。进一步地，对于带噪声样本，我们的界描述了依赖于逼近空间$V_m$维度$m\)的偏差-方差权衡。所有结果均以高概率成立。为进行演示，我们考虑了定义在$d$维立方体上并通过均匀随机样本给出的函数。我们分析了多项式、半周期余弦，以及非周期Sobolev空间$H_{\mathrm{mix}}^2$的有界正交基。克服该$H_{\text{mix}}^2$基的数值问题，这给出了一种具有二次误差衰减的新型稳定逼近方法。数值实验表明了我们的结果的适用性。